Isoparametric elements

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Isoparametric elements For elements with complicated geometrical shape like curved sides or surfaces in 3D it is advantageous to transform the geometry from cartesian to natural, curvilinear coordinates ζ, η. Fig.5-1 Transformation of isoparametric element from cartesian to natural coordinates

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Isoparametric elements

For elements with complicated geometrical shape like curved sides or surfaces in 3D it is advantageous to transform the geometry from cartesian to natural, curvilinear coordinates ζ, η.

Fig.5-1 Transformation of isoparametric element from cartesian to natural coordinates

The transformation relations x = x(ζ, η), y = y(ζ, η), can be formulated in a similar form as the displacement approximation. If the displacements are approximated as

and the coordinates transformed according to

Then, in case of the element is called isoparametric. The reason is that we need the same number of deformation parameters ui, vi to describe displacement field as the number of nodal coordinates xi, yi to describe element geometry.

Shape functions of isoparametric elements can be described in a systematic way as families of certain type and we shall mention two of such types - families with linear and quadratic shape functions.

ISOPARAMETRIC ELEMENTS WITH LINEAR SHAPE FUNCTIONS

1D elements

Linear shape functions expressed in natural coordinate over a unit element

Nζ1 = ( 1 – ζ ) / 2 , Nζ2 = ( 1 + ζ ) / 2

are plotted in Fig.5-2

Fig.5-2 Linear shape functions

We note that they are the shape functions of LINK1 and LINK8 elements, described previously.

2D elements - bilinear element

Fig.5-3 Bilinear element in cartesian and natural coordinates

Bilinear element in Fig.5-3 has shape functions generated by multiplying linear expressions in ζ and η direction. Due to multiplication, the product is not a linear function: N1 = Nζ1 . Nη1 = (1 - ζ).(1 - η ) / 4

N2 = Nζ2 . Nη1 = (1 + ζ).(1 - η ) / 4

N3 = Nζ2 . Nη2 = (1 + ζ).(1 + η ) / 4

N4 = Nζ1 . Nη2 = (1 - ζ).(1 + η ) / 4

One of the shape functions is shown in Fig.5-4a). Besides the basic isoparametric functions, quadratic „extra“ shape functions are sometimes used to improve the quality of bilinear element.

Fig.5-4 Isoparametric a) and extra b), c) shape functions of bilinear element

In ANSYS, this quadrilateral element can be found under the names PLANE42, or PLANE182. As mentioned, triangular form of this element can be used, too.

Examples 5.1 and 5.2 show improved quality of the bilinear element with extra shapes in comparison to a standard bilinear, and especially to triangular element.

3D elements with linear functions in ζ, η and ξ

Eight shape functions of the hexaedral element according to Fig.5-6 a) are created by mutual multiplication of linear functions in ζ, η and ξ :

They are often complemented by extra quadratic shape functions, improving the element properties :

Besides the hexaedral shape, other „degenerated“ element shapes ACCORDING TO Fig5-6 b),c),d) can be obtained. In some FE systems such elements are classified as special types, in ANSYS they are all seen as degenerated shapes of the same element SOLID45 or SOLID 185.

Fig.5-6 Eight node hexaedral element and its degenerated shapes

The shape 5-6 a) is used to create mapped meshes, which are saving computer time and memory, but are difficult to prepare. The shape 5-6 d) is used for fully automatic free meshing, and the intermediate shapes b), c) can be used for transition between these two mesh types.

ISOPARAMETRIC ELEMENTS WITH QUADRATIC SHAPE FUNCTIONS

1D elements

Quadratic shape functions expressed in natural coordinate over an unit element

are plotted in Fig.5-7

Fig.5-7 Quadratic 1D element and its shape functions

By a systematic evolution of quadratic elements, similar to the evolution of linear elements we obtain 9-node plane element according to Fig.5-8 a). Its nine shape functions are obtained by multiplication of qudratic functions from the preceding paragraph: N1 = 2Nζ1 .2Nη1, N2 = 2Nζ2 .2Nη1, N3 = 2Nζ3 .2Nη1,

N4 = 2Nζ3 .2Nη2, N5 = 2Nζ3 .2Nη3, N6 = 2Nζ2 .2Nη3,

N7 = 2Nζ1 .2Nη3, N8 = 2Nζ1 .2Nη2, N9 = 2Nζ2 .2Nη2,

Practical usage of this element is limited and the 8-node version according to Fig.5-8 b) prevails. In ANSYS, degenerated triangular form with curved sides (Fig. 5-8 c) is included in the same element type PLANE82, or PLANE183.